Since the set is large we will look only at forming squares
with an enhanced set. We omit the heptomino with a hole and so need an extra
five copies of one of the pieces to form a side 28 square.

For those pieces which form a 2x7 rectangle we can form squares
of all sizes. The diagram below shows side 28 and 35 squares.

Since any 7 x even rectangle is possible with these pieces any
square can be increased in size by 14. Also these pieces can also form a 21x21
square which will allow squares to be increased in size by 21 thus showing
that all squares of side 7k (k>3) can be made.

The remaining pieces can all be used to make
squares of side 28. The diagrams are here.

A number of other pieces can also form a side 35 square. These
pieces will prbably also form larger squares.